Question
Question
Which expression is correctly developed to use the Power Rule of Exponents?(1 point)
Responses
(78)⋅(710)
left parenthesis 7 superscript 8 baseline right parenthesis times left parenthesis 7 superscript 10 baseline right parenthesis
129
12 superscript 9 baseline
(74)8
left parenthesis 7 superscript 4 baseline right parenthesis superscript 8 baseline
(−5)0
Answers
GPT-4o mini
To determine which expression is correctly developed to use the Power Rule of Exponents, let's review the options provided:
1. **(78)⋅(710)** - This represents \(7^8 \cdot 7^{10}\). According to the Power Rule of Exponents, when multiplying two expressions with the same base, you add the exponents: \(7^8 \cdot 7^{10} = 7^{8+10} = 7^{18}\).
2. **129** - This is simply the number 129 and does not involve exponents.
3. **(74)8** - This represents \((7^4)^8\). According to the Power Rule for exponents, when raising a power to another power, you multiply the exponents: \((7^4)^8 = 7^{4 \cdot 8} = 7^{32}\).
4. **(−5)0** - This expression states that any non-zero number raised to the power of 0 equals 1, so \((-5)^0 = 1\). While true, it does not illustrate the use of the Power Rule with exponents in the same sense as the first two options.
The two expressions that illustrate the Power Rule of Exponents are the first and third options. However, if we are strictly looking for a situation where the Power Rule is applied directly (like multiplying powers with the same base or raising a power to a power), we can choose:
- **(78)⋅(710)** for the case of multiplying the same base (though both yield correctly to another form).
- **(74)8** clearly reflects raising a power to another power.
In conclusion, both (1) and (3) illustrate the application of exponent rules, but if we select one for the specific development context:
**(74)8** is a clear application of raising to a power, converting to \(7^{32}\).
1. **(78)⋅(710)** - This represents \(7^8 \cdot 7^{10}\). According to the Power Rule of Exponents, when multiplying two expressions with the same base, you add the exponents: \(7^8 \cdot 7^{10} = 7^{8+10} = 7^{18}\).
2. **129** - This is simply the number 129 and does not involve exponents.
3. **(74)8** - This represents \((7^4)^8\). According to the Power Rule for exponents, when raising a power to another power, you multiply the exponents: \((7^4)^8 = 7^{4 \cdot 8} = 7^{32}\).
4. **(−5)0** - This expression states that any non-zero number raised to the power of 0 equals 1, so \((-5)^0 = 1\). While true, it does not illustrate the use of the Power Rule with exponents in the same sense as the first two options.
The two expressions that illustrate the Power Rule of Exponents are the first and third options. However, if we are strictly looking for a situation where the Power Rule is applied directly (like multiplying powers with the same base or raising a power to a power), we can choose:
- **(78)⋅(710)** for the case of multiplying the same base (though both yield correctly to another form).
- **(74)8** clearly reflects raising a power to another power.
In conclusion, both (1) and (3) illustrate the application of exponent rules, but if we select one for the specific development context:
**(74)8** is a clear application of raising to a power, converting to \(7^{32}\).
Which of the following correctly used the Power Rule of Exponents in the expression (109)7 ?(1 point)
Responses
1016
10 superscript 16 baseline
102
10 superscript 2 baseline
1063
10 superscript 63 baseline
907
Responses
1016
10 superscript 16 baseline
102
10 superscript 2 baseline
1063
10 superscript 63 baseline
907
GPT-4o mini
To analyze the expression \((10^9)^7\) in terms of the Power Rule of Exponents, we apply the rule that states \((a^m)^n = a^{m \cdot n}\). In this case, we have:
\[
(10^9)^7 = 10^{9 \cdot 7} = 10^{63}
\]
Now let's examine the provided response options:
1. **1016** - This represents \(10^{16}\), which is incorrect.
2. **102** - This represents \(10^{2}\), which is incorrect.
3. **1063** - This represents \(10^{63}\), which is correct.
4. **907** - This represents \(9^{7}\), which is also incorrect.
The correct response that rightly uses the Power Rule of Exponents in the expression \((10^9)^7\) is:
**1063** (which corresponds to \(10^{63}\)).
\[
(10^9)^7 = 10^{9 \cdot 7} = 10^{63}
\]
Now let's examine the provided response options:
1. **1016** - This represents \(10^{16}\), which is incorrect.
2. **102** - This represents \(10^{2}\), which is incorrect.
3. **1063** - This represents \(10^{63}\), which is correct.
4. **907** - This represents \(9^{7}\), which is also incorrect.
The correct response that rightly uses the Power Rule of Exponents in the expression \((10^9)^7\) is:
**1063** (which corresponds to \(10^{63}\)).
If the expression 410 was simplified using the Power Rule of Exponents, what is an equivalent expanded expression? (1 point)
Responses
(45)2
left parenthesis 4 superscript 5 baseline right parenthesis squared
(412)2
left parenthesis 4 superscript 12 baseline right parenthesis squared
4⋅4⋅4⋅4⋅4⋅4⋅4
4 times 4 times 4 times 4 times 4 times 4 times 4
(42)8
Responses
(45)2
left parenthesis 4 superscript 5 baseline right parenthesis squared
(412)2
left parenthesis 4 superscript 12 baseline right parenthesis squared
4⋅4⋅4⋅4⋅4⋅4⋅4
4 times 4 times 4 times 4 times 4 times 4 times 4
(42)8
GPT-4o mini
To simplify the expression \(4^{10}\) using the Power Rule of Exponents, we can look for equivalent expanded expressions.
Let’s break down the options:
1. **(45)2** - This represents \((4^5)^2\). Applying the Power Rule: \((4^5)^2 = 4^{5 \cdot 2} = 4^{10}\). This is correct and simplifies to \(4^{10}\).
2. **(412)2** - This represents \((4^{12})^2\). Applying the Power Rule: \((4^{12})^2 = 4^{12 \cdot 2} = 4^{24}\). This is incorrect.
3. **4⋅4⋅4⋅4⋅4⋅4⋅4** - This represents \(4^{7}\) (since there are 7 factors of 4), which is incorrect.
4. **(42)8** - This represents \((4^2)^8\). Applying the Power Rule: \((4^2)^8 = 4^{2 \cdot 8} = 4^{16}\). This is also incorrect.
The only option that simplifies and represents \(4^{10}\) correctly is:
**(45)2**, which corresponds to \((4^5)^2\).
Let’s break down the options:
1. **(45)2** - This represents \((4^5)^2\). Applying the Power Rule: \((4^5)^2 = 4^{5 \cdot 2} = 4^{10}\). This is correct and simplifies to \(4^{10}\).
2. **(412)2** - This represents \((4^{12})^2\). Applying the Power Rule: \((4^{12})^2 = 4^{12 \cdot 2} = 4^{24}\). This is incorrect.
3. **4⋅4⋅4⋅4⋅4⋅4⋅4** - This represents \(4^{7}\) (since there are 7 factors of 4), which is incorrect.
4. **(42)8** - This represents \((4^2)^8\). Applying the Power Rule: \((4^2)^8 = 4^{2 \cdot 8} = 4^{16}\). This is also incorrect.
The only option that simplifies and represents \(4^{10}\) correctly is:
**(45)2**, which corresponds to \((4^5)^2\).
Use the Power Rule of Exponents to create an expression equivalent to (156)2.(1 point)
Responses
1512
15 superscript 12 baseline
154
15 superscript 4 baseline
153
15 cubed
158
Responses
1512
15 superscript 12 baseline
154
15 superscript 4 baseline
153
15 cubed
158
GPT-4o mini
To create an expression equivalent to \((15^6)^2\) using the Power Rule of Exponents, we apply the rule that states \((a^m)^n = a^{m \cdot n}\).
So we have:
\[
(15^6)^2 = 15^{6 \cdot 2} = 15^{12}
\]
Now, let's evaluate the provided response options:
1. **1512** - This represents \(15^{12}\), which is correct.
2. **154** - This represents \(15^4\), which is incorrect.
3. **153** - This represents \(15^3\), which is incorrect.
4. **158** - This represents \(15^8\), which is incorrect.
The equivalent expression that correctly applies the Power Rule of Exponents to \((15^6)^2\) is:
**1512** (which corresponds to \(15^{12}\)).
So we have:
\[
(15^6)^2 = 15^{6 \cdot 2} = 15^{12}
\]
Now, let's evaluate the provided response options:
1. **1512** - This represents \(15^{12}\), which is correct.
2. **154** - This represents \(15^4\), which is incorrect.
3. **153** - This represents \(15^3\), which is incorrect.
4. **158** - This represents \(15^8\), which is incorrect.
The equivalent expression that correctly applies the Power Rule of Exponents to \((15^6)^2\) is:
**1512** (which corresponds to \(15^{12}\)).
What exponent would make the two expressions equivalent: (33)x=324?(1 point)
Responses
21
21
15
15
8
8
27
Responses
21
21
15
15
8
8
27
GPT-4o mini
To determine the value of \( x \) that makes the two expressions \( (3^3)^x = 324 \) equivalent, we first need to convert the number 324 into a base of 3.
Let's start by simplifying the left-hand side:
\[
(3^3)^x = 3^{3x}
\]
Next, we need to express 324 as a power of 3. We can factor 324:
\[
324 = 3^4 \times 4 = 3^4 \times 2^2
\]
Since \( 2^2 \) is not a power of 3, let's focus on the direct powers of 3.
Calculating \( 3^1\), \( 3^2\), \( 3^3\), and \( 3^4\):
- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 27 \)
- \( 3^4 = 81 \)
- \( 3^5 = 243 \)
- \( 3^6 = 729 \)
Neither \( 3^4 \) nor \( 3^5 \) gives us 324. So, let's directly divide:
\[
324 \div 3 = 108 \\
108 \div 3 = 36 \\
36 \div 3 = 12 \\
12 \div 3 = 4 \\
4 \div 2 = 2 \\
2 \div 2 = 1
\]
At this point, we realize \( 324 = 3^4 \times 2^2 \) doesn't fit neatly into our exponent.
Instead, let's just compute:
\[ x = \frac{\log(324)}{\log(27)} \]
Since we know that \( 27 = 3^3 \):
\[
3^{3x} = 3^4 \times 4
\]
Using \( 3^6 \):
From calculations,
324 = 3^4 \times 2^2 is not an equivalently achieving power; but by close estimates we can search nearby:
- \( 3^4 < 324 < 3^5 \)
Let's check with:
So we can set up the equations:
\[
3^{3x} = 324
\]
Now equate:
3x = log(324) / log(3)
Solving shows a disorder and proves complicated.
But if we reshape only exponent handle:
Solving simplified becomes:
\[
3x = 4 => x = 4/3 = 1.333
\]
So among choices gained reveals no clear matches.
To conclude:
None of the provided options \( 21, 15, 8, 27 \) directly yield closeness to exponent versatility targeting closest matched equivalent.
Conclusively, the answer is **not** among provided options but needs re-evaluative airporting claiming upwards to better details seen.
If requested an immediate match, the closest to fulfill calculation aspect might **not** find closure aligning best from recurrence hints remaining.
Let's start by simplifying the left-hand side:
\[
(3^3)^x = 3^{3x}
\]
Next, we need to express 324 as a power of 3. We can factor 324:
\[
324 = 3^4 \times 4 = 3^4 \times 2^2
\]
Since \( 2^2 \) is not a power of 3, let's focus on the direct powers of 3.
Calculating \( 3^1\), \( 3^2\), \( 3^3\), and \( 3^4\):
- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 27 \)
- \( 3^4 = 81 \)
- \( 3^5 = 243 \)
- \( 3^6 = 729 \)
Neither \( 3^4 \) nor \( 3^5 \) gives us 324. So, let's directly divide:
\[
324 \div 3 = 108 \\
108 \div 3 = 36 \\
36 \div 3 = 12 \\
12 \div 3 = 4 \\
4 \div 2 = 2 \\
2 \div 2 = 1
\]
At this point, we realize \( 324 = 3^4 \times 2^2 \) doesn't fit neatly into our exponent.
Instead, let's just compute:
\[ x = \frac{\log(324)}{\log(27)} \]
Since we know that \( 27 = 3^3 \):
\[
3^{3x} = 3^4 \times 4
\]
Using \( 3^6 \):
From calculations,
324 = 3^4 \times 2^2 is not an equivalently achieving power; but by close estimates we can search nearby:
- \( 3^4 < 324 < 3^5 \)
Let's check with:
So we can set up the equations:
\[
3^{3x} = 324
\]
Now equate:
3x = log(324) / log(3)
Solving shows a disorder and proves complicated.
But if we reshape only exponent handle:
Solving simplified becomes:
\[
3x = 4 => x = 4/3 = 1.333
\]
So among choices gained reveals no clear matches.
To conclude:
None of the provided options \( 21, 15, 8, 27 \) directly yield closeness to exponent versatility targeting closest matched equivalent.
Conclusively, the answer is **not** among provided options but needs re-evaluative airporting claiming upwards to better details seen.
If requested an immediate match, the closest to fulfill calculation aspect might **not** find closure aligning best from recurrence hints remaining.